The Hamiltonian for an ionic crystal lattice in a uniform external magnetic field
[1]}, [2]}, [3]} can be written in a compact form as
\(H & = &\frac{1}{2} (p-{ \tilde{A} }u)^T (p-{ \tilde{A} }u) + \frac{1}{2} u^T K u \qquad \nonumber \\& = &\frac{1}{2} p^T p + \frac{1}{2} u^T (K-{ \tilde{A} }^{2}) u + u^T\!{ \tilde{A} }\,p.\) 
